Objective Function

The objective of this model is minimizing the operational costs of a subset of cooperating terminal operating companies. Thereby, the operational costs are defined as discussed in Section 2.3, as the costs that are subject to change when cooperating.

In line with standard notation in game theory, the term coalition will be used in this model to refer to a subset of cooperating TOCs. This means that when a coalition is denoted by S and the set of terminal operating companies of Brabant Intermodal B.V. is denoted by N, it holds that

S N. Thereby, N = {1, 2, 3, O} (with n  1=ITV, 2=ROCW, 3=BTT and O=OCT). Within this set of TOCs, a set of satellite terminals T = {1, 2, 3} (with t  1=ITV, 2=ROCW, 3=BTT), and a set of internal hub terminals H = {O} (with h  O = OCT) can be distinguished. This

19

means that n can either be a satellite terminal (t) or a hub terminal (h). Furthermore, there is an external hub terminal denoted by M.

The operational costs of a coalition (SN) is defined as OC_S. Hence, the mathematical objective function of the MILP is:

min OC_s

The operational costs of a cooperation are composed of the operational costs (trucking, barging and handling) of the satellite terminals (t) and the hub terminals (h) that are partner of a coalition and the costs for these terminals of involving the external partner.

,

The symbols as used in the total operational cost function are explained in Table 6.

Table 6: Definition of the symbols used in the Total Operational Cost function.

,

BCt S The direct barging costs of satellite terminal t under coalition S.

, , t O S

BC The indirect barging costs via OCT of satellite terminal t under coalition S.

, , t M S

BC The indirect barging costs via an external partner of satellite terminal t under coalition S.

, ,

t rent S

BC The costs for satellite terminals for renting the barges per week under coalition S.

,

TCt S The direct trucking costs of satellite terminal t under coalition S.

, , t O S

TC The indirect trucking costs via OCT of satellite terminal t under coalition S.

, , t M S

TC The indirect trucking costs via an external partner of satellite terminal t under coalition S.

,

STCt S The trucking costs of satellite terminal t under coalition S for transporting the barging volumes from the customers to the terminal.

,

HCt S The handling costs of satellite terminal t under coalition S.

,

BCO S The direct barging costs of hub terminal OCT under coalition S.

, ,

O rent S

BC The costs for hub terminal OCT for renting the barges per week under coalition S.

,

TCO S The direct trucking costs of hub terminal OCT under coalition S.

,

STCO S The trucking costs of hub terminal OCT under coalition S for transporting the barging volumes from the customers to the terminal.

,

HCO S The handling costs of hub terminal OCT under coalition S.

,

ECM S The external partner costs under coalition S.

The formulas used to determine the values in the total cost function per terminal are defined in Appendix G. The costs for the individual terminals will be summed in the total cost function above over all terminals participating in coalition S.

The structure of the cost functions will be discussed in this section. Thereby one of the formulas of each cost component will be given as an example. As the formulas for the various types of terminals (satellites, hubs and external partners) and links are only slightly different, these are given in Appendix G.

20 3.3.1.1 Barging Costs

The costs for barging are composed of two components, the costs for renting a barge and the fuel costs.

There exist two types of renting agreements, the renting agreements per week and the renting agreements per trip. The first type of agreement involves a contract for a whole year with a charged price per week, independent of the number of trips that is made within that week.

Hence, when the number of trips of such a barge in a year is larger than zero, the price per week needs to be paid for the whole year. The second type of agreement, the agreements per trip, charges a price per trip. For these barges it matters whether a barge is used on a link to the port or on a link to a hub terminal. OCT and ROCW both have renting agreements per week for their barges, ITV and BTT have renting agreements per trip.

Besides these costs for renting a barge, costs are involved for the fuel consumption. Some contracts include a component fuel costs. In these types of contracts, the fuel consumption is included until a certain limit of the fuel price. This means that when the fuel price is 500 euro/m³ and the limit of the price is 300 euro/m³, 200 euro/m³ needs to be paid extra for each m³ fuel consumed on a trip.

This results for example in the following cost functions for barging between a satellite terminal and a port terminal:

In the first cost function the number of trips of a barge on a link is multiplied by the barge price per trip of that barge on the specific link and the number of trips of a barge on a link is multiplied by the fuel consumption on that link times the fuel price minus the included component of the fuel price (x_b). In these functions, it holds that for each barge either p_b,tp=0 or p_b,w = 0.

In the second cost function, the renting costs for the barges with agreements per week are calculated. A binary variable y_b is used to resemble whether a barge is used or not.

In Table 26 of Appendix G, the cost functions related to barging costs are summarized; these costs functions are given in a modular way as how they are defined in the AIMMS program. This modular way of defining the costs makes it easier to check the formulas and the correctness of the model calculations.

3.3.1.2 Trucking Costs

Six trucking cost functions are defined (Table 27 of Appendix G). These cost functions are related to direct trucking, indirect trucking and short-haul trucking. The short-haul trucking costs are involved for trucking between a terminal t and its customers. These costs occur when something is either directly or indirectly barged to the terminal. When a container is transported indirectly by truck, the trucking between the customer and the hub terminal is seen as indirect trucking and not as a short-haulage.

The formula for transporting between a customer and the port directly is:

21

In this formula a variable cost component per kilometer is multiplied by two times the distance between two locations. It is multiplied by two as it is assumed that there is no re-use. Hence, the route that needs to be travelled to transport one or two TEU is resembled by trucking from the port to the customer and a return to the port to bring the container back. The second component in these cost functions resembles the variable cost per hour. This variable cost component is multiplied by the sum of the travel time, the waiting time at the relevant locations and the average delay on the route between two locations. The travel time is determined by dividing two times the distance by the average speed on a route. The average speed will be set higher on a larger distance than on a smaller distance. The average delay, due to traffic jams, accidents and other external factors also depends on the distance that needs to be travelled. Finally, both costs components are multiplied by the number of trucking that is required. This number of trucking depends on the number of TEU transported per barge and the average amount of TEU transported on a truck.

3.3.1.3 Handling Costs

Another component of the operational costs are the handling costs. Based on Van Rooy (2010) the handling costs can be divided in three categories; the fixed, semi-fixed and variable costs. The fixed costs will not change when the number of handlings increases. The semi-fixed costs are costs that will increase when a certain handling volume is reached; it is a gradually increasing cost function. Finally, the variable costs part increases per extra handling. In this case, it is likely that an investment is needed when OCT is part of the coalition and the terminals transport via OCT.

To deal with this cost, the semi-fixed cost cannot be assumed to be constant. It can be that an increase in handling volumes at these terminals requires extra FTE hours for the crane operations, extra FTE hours for planning and the extra coverage of fixed costs. In this model, these semi-fixed costs are included partly as fixed costs and partly as variable costs that are linear with the number of handlings. Only the variable costs of handling are included in the model, since the fixed costs of handling will not change as a consequence of cooperation. An overview of the fixed, semi-fixed and variable costs is given in Appendix H.

The standard cost function for handling at a satellite terminal is (Table 28 of Appendix G):

,

The number of handlings at a satellite terminal is related to the volume that is transported per barge either directly or indirectly. The volume that is transported per barge needs to be divided by the TEUF factor and multiplied by two to determine the required number of handlings. The number of handlings at a satellite terminal depends on the volumes that are send from the satellite terminals either by barge or by truck and the own volumes of the hub terminal transported by barge.

3.3.1.4 Costs of External Partner

As discussed in Chapter 1, it is expected that terminals can transport via an external partner. This external partner charges a price for its service. This price is independent of the end destination in the port and includes the barging and the handling of the volumes. The price for handling is charged as a price per container handling and the price for barging as a price per TEU.

22

In Table 30 of Appendix G, the demand flow related decision variables are given. The model has eight types of demand flow related decision variables, since there are six possible paths for each satellite terminal and two possible paths for each hub terminal. Related to these demand flow related decision variables, the model has four types of decision variables regarding the number of trips that are made on the specific links (Table 31 of Appendix G). For the satellite terminals exists three possible barging paths on which needs to be decided about the number of trips of each barge (direct, indirect via OCT, indirect via external hub). Furthermore, the hub OCT has one connection on which needs to be decided about the number of trips of each barge. Finally, a binary type of decision variable is included, which resembles the decision of whether a barge is operated or not (Table 32 of Appendix G). This binary decision variable is needed for calculating the costs of the barges that are rented per week.

3.3.3 Parameters

A summary of all input parameters is given in Appendix G (Table 33 to Table 42).

One category of input parameters is related to the demand information of each terminal (Table 33). This demand depends on the coalition and its expected volume growth as a consequence of an improved reliability. Furthermore, it depends on the natural growth. The demand input parameters in the projected year are determined by increasing the demand of the reference year (2010) by the expected total demand growth. The other two parameters in this category are related to the percentages of rush orders. These percentages depend on the coalition in which is cooperated. The demand of the reference year 2010 is given in Appendix I.

The other parameters are all related to the relevant costs that need to be determined. These can be divided in four categories: barging parameters, trucking parameters, handling parameters and external partner parameters.

For the determination of the first category of relevant costs, the barging costs, a variety of parameters is needed. As is given in Section 3.3.1.1, the barging costs depend on the prices for barges per week, the prices for barges per trip, the fuel consumption per trip and the fuel prices.

The definitions of these parameters are all given in Table 34 of Appendix G. The values of these parameters are based on a previous research by Van Rooy (2010), questionnaires filled in by the representatives of the various terminals and the company supervisors. The values are given in Appendix J.

Furthermore, a decision variable is included in the cost function for barging, namely the number of trips of the barges on the various connections. This decision variable is influenced by the capacity, the operating time of barges per year, the maximum feasible utilization rates and the throughput time of barging. These parameters are all defined in Table 35 and Table 36 of Appendix G. A maximum feasible utilization rate of the barges needs to be set since the utilization rates of some terminals are restricted by exogenous factors. For example on one of the barges of ITV 28 TEU can be transported. However, part of this 28 TEU barge can only be filled

In document Eindhoven University of Technology MASTER The determination and division of benefits among partners of a horizontal cooperation in transportation Soons, D.E. (pagina 31-35)